A Many-valued Semantics for Multi-agent System
Yang Song and Satoshi Tojo
Japan Advanced Institute of Science and Technology, Japan
Keywords:
Weak Kleene Logic, Strong Kleene Logic, Many-valued Logic, Multi-agent System, Agent Communication.
Abstract:
We often employ epistemic logic to express the epistemic states of agents. However, it is often too complicated
to build a Kripke model because we should consider all possibilities of the knowledge between agents. In this
paper, we employ a many-valued logic to express the epistemic states of agents. Thus far, the representations
usually show the epistemic state of a single agent, however, we apply the logic to the multi-agent system.
Here, we consider that there exist three kinds of epistemic states of known, truth-value unknown, and content
unknown. Furthermore, we introduce two kinds of agent communication in our semantics, i.e., teaching and
asking, and show how the epistemic states of agents will change.
1 INTRODUCTION
We often employ epistemic logic to express the epis-
temic states of agents, e.g.,
a
A stands for agent
a knows that A is true and ¬
a
B stands for that a
doesn’t know that B is true. Actually, if we build the
Kripke model perfectly and consider that every agent
has the common sense which is shown in the model,
the representation of epistemic states and the simula-
tion of agent communication usually work very well.
However, it is often too complicated to show the epis-
temic states of multi-agent system in the modal logic
because we should consider all possibilities of knowl-
edge between agents, e.g., agent a knows that b knows
p while agent b doesn’t know that agent a knows that
agent b knows p.
Instead of epistemic logic, in this research, we
consider employing many-valued logic to avoid those
notational complications. In the 3-valued logic,
we can use the third value, which can be read
as “unknown” or “unknown whether true or
false”(Kleene, 1952) to show such a state. (Ciucci
and Dubois, 2012) provided a translation from the
strong Kleene logic to epistemic logic. (Szmuc, 2019)
considered the third value in the paraconsistent weak
Kleene logic as an epistemic interpretation. In the 4-
valued logic, the four values are usually called true,
false, neither and both. Belnap considered the val-
ued as follows(Belnap, 1977):
• The value of p is true(T) means that the computer
is told that p is true.
• The value of p is false(F) means that the computer
is told that p is false.
• The value of p is neither(N) means that the com-
puter is not told anything about p.
• The value of p is both(B) means that the computer
is told that p is both true and false (perhaps from
different sources, or so on).
If we consider the computer as an agent, the 4 val-
ues can be seen as the epistemic states of the agent,
i.e., true(T) and false(f) mean that p is known to the
agent, while neither(N) and both(B) mean that p is
unknown to the agent.
However, both of the representations only show
the epistemic state of a single agent, while it can be
the case that a certain proposition is known to some
agents while unknown to others in a multi-agent sys-
tem. In this paper, we give a new many-valued logic
semantics to express the epistemic states in the multi-
agent system. Here, we consider that each proposition
is either true or false as the classical logic, while we
add several additional values to show the epistemic
states of agents. Therefore, all of the propositions
have the same classical values for each agent, while
the epistemic states is different between the agents.
This paper is organized as follows. In Section
2, we show a fundamental of many-valued seman-
tic consequence relation and introduce some many-
valued logics. In Section 3, we give two pair seman-
tics for two readings of knowledge. In Section 4, we
combine the two semantics and extend it to express
the epistemic states of multi-agent system. In Section
5, we introduce two kinds of agent communication in
our semantics. Finally in Section 6, we conclude.
Song, Y. and Tojo, S.
A Many-valued Semantics for Multi-agent System.
DOI: 10.5220/0010918300003116
In Proceedings of the 14th International Conference on Agents and Artificial Intelligence (ICAART 2022) - Volume 3, pages 863-870
ISBN: 978-989-758-547-0; ISSN: 2184-433X
Copyright
c
2022 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
863
2 MANY-VALUED LOGIC
In the many-valued logic, we usually consider that the
propositional language L consists of a set {∼, ∧, ∨}
of propositional connectives and a countable set Prop
of propositional variables. We denote by Form the set
of formulas defined as usual in L , denote a formula
of L by A, B, C, etc. and a subset of Form by Γ, ∆, Σ,
etc.
Here, we show a fundamental of many-valued
semantic consequence relation(Szmuc and Omori,
2018) we use in this paper.
Definition 1 (Univalent semantics). A univalent se-
mantics for the language L is a structure M =
hV , D, δi, where
• V is a non-empty set of truth values,
• D is a non-empty proper subset of V ,
• δ contains, for every n-ary connective ∗ in the lan-
guage, a truth-function δ
∗
: V
n
→ V .
A univalent interpretation is a pair hM, µi, where M is
such a structure, and µ is an evaluation function from
Prop to V . Given an interpretation, µ is extended to
a map from Form to V recursively, by the following
clause:
• µ(∗(A
1
, . . . , A
n
)) = δ
∗
(µ(A
1
), . . . , µ(A
n
)).
Finally, Γ |=
M
A iff for all univalent interpretation
hM, µi, if µ(B) ∈ D for all B ∈ Γ, then µ(A) ∈ D.
Semantically speaking, V shows the possible val-
ues in a logic, D shows the designated values, and δ
shows the truth tables of the connectives in a logic.
By this definition, we can give semantic conse-
quence relations for many-valued logics.
• The semantic consequence relation |=
CL
for clas-
sical propositional logic is obtained by setting
V = {t, f}, D = {t}.
• In the strong Kleene logic and logic of paradox, if
we write the third value as b , then the truth table
is written as following:
∼
t f
b b
f t
∧ t b f
t t b f
b b b f
f f f f
∨ t b f
t t t t
b t b b
f t b f
– The semantic consequence relation |=
SK
for
strong Kleene logic is obtained by setting V =
{t, b, f}, D = {t}.
– The semantic consequence relation |=
LP
for
logic of paradox is obtained as above except
that we replace D = {t} by D = {t, b}.
Normally, b can be read as unknown, undecided,
etc.
• In the weak Kleene logic, if we write the third
value as n , then the truth table is written as fol-
lowing:
∼
t f
n n
f t
∧ t n f
t t n f
n n n n
f f n f
∨ t n f
t t n t
n n n n
f t n f
– The semantic consequence relation |=
WK
for
weak Kleene logic (WK) is obtained by setting
V = {t, n, f}, D = {t}.
– The semantic consequence relation |=
PWK
for
paraconsistent weak Kleene logic (PWK) is
obtained as in WK except that we replace D =
{t} by D = {t, n}.
Normally, n can be read as meaningless, unde-
fined, off-topic, etc.
From the truth table, we can see that the value of a
formula will be n even if one atom of the formula
has the value n. Therefore, the weak Kleene logic
is also called infectious logic and n is called an
infectious value.
• Belnap considered a 4-valued logic called first-
degree entailment logic FDE. The four values are
usually written as {t, f, b, n}. The truth table is as
following(Omori and Wansing, 2017):
∼
t f
b b
n n
f t
∧ t b n f
t t b n f
b b b f f
n n f n f
f f f f f
∨ t b n f
t t t t t
b t b t b
n t t n n
f t b n f
Deutsch considered another 4-valued logic that is
usually called that is called S
f de
logic. The read-
ing of values is the same as FDE and the truth
table is as following(Ferguson, 2017):
∼
t f
b b
n n
f t
∧ t b n f
t t b n f
b b b n f
n n n n n
f f f n f
∨ t b n f
t t t n t
b t b n b
n n n n n
f t b n f
The semantic consequence relations of |=
FDE
and
|=
Sfde
are both obtained by setting V = {t, f, b, n},
D = {t, b}. Also, it is easy to see from the truth
table that the S
f de
logic can also be considered as
the combination of the logic of paradox and the
weak Kleene logic.
3 PAIR SEMANTICS FOR
MULTI-AGENT SYSTEM
We usually use the epistemic logic to show the knowl-
edge of a multi-agent system. Normally, we can use
ICAART 2022 - 14th International Conference on Agents and Artificial Intelligence
864
the formula ¬
a
A ∧ ¬
a
¬A to express the fact that
A is unknown to agent a. However, sometimes the
Kripke model of epistemic logic is too complex, so
here we give a simple way to express the states of
agent knowledge by many-valued logic.
We define the classical valuation V
t
: Prop → {t, f}
as usual. Then we define the knowledge of agent a as
the valuation V
a
: Prop → {1, 0}.
• V
t
(p) = t means p is true.
• V
t
(p) = f means p is false.
• V
a
(p) = 1 means p is known to agent a.
• V
a
(p) = 0 means p is unknown to agent a.
Before showing the semantics, we want to ask the
question that What is knowledge?. To show this
question clearly, consider the following situation,
Example 2. p, q are two propositions and p is known
to agent a while q is unknown. Assume that p is true,
is the formula p ∨ q known to a?
Agent a can acquire the knowledge that p ∨ q is
true. Actually, it is the same in the epistemic logic
that p → (p ∨ q) no matter what is q. Can we say
that p ∨ q is known to a?
Here, we show two semantics for two readings of
knowledge.
• Semantics (I) stands for the case that we consider
that p ∨ q is unknown to a. In other words, we
consider that a formula A is known to a agent if
and only if all of the atoms of A are known to the
agent. Therefore, the state unknown is infectious
and the semantics should be similar with that of
weak Kleene logic which we introduce in Section
2.
• Semantics (II) stands for the case that we consider
that p ∨ q is known to a. In other words, we con-
sider that a formula A is known to a agent if and
only if the agent knows whether A is true or not.
Therefore, the semantics should be similar with
that of epistemic logic.
3.1 Semantics (I)
If we consider the state of unknown as the first case,
we should give a semantics for the infectious logic.
Actually, we have already shown a pair semantics to
express the infectious logic in(Song et al., 2021).
Definition 3. A two-valued interpretation for the lan-
guage L is a pair hV
w
t
, V
w
a
i, where V
w
t
: Prop → {t, f}
and V
w
a
: Prop → {0, 1}. Valuations V
w
t
, V
w
a
are then
extended to interpretations I
t
, I
m
by the following con-
ditions.
• I
w
t
(p)=t iff V
w
t
(p)=t
• I
w
a
(p)=1 iff V
a
(p)=1
• I
w
t
(∼A)=t iff I
w
t
(A)=f
• I
w
a
(∼A)=1 iff I
w
a
(A)=1
• I
w
t
(A ∧ B)=t iff I
w
t
(A)=t and I
w
t
(B)=t
• I
w
a
(A ∧ B)=1 iff I
w
a
(A)=1 and I
w
a
(B)=1
In this paper, we consider that A∨B as the same as
∼ (∼ A∧ ∼ B). If we read the additional value V
w
a
as
unknown, it can be considered as the semantics that
shows the epistemic states of agents.
Definition 4. A four-valued interpretation of L is a
function I
w
4
: Prop → {t1, t0, f0, f1}. Given a four-
valued interpretation I
s
4
, this is extended to a function
that assigns every formula a truth value by the follow-
ing truth functions:
A
∼ A
t1 f1
t0 f0
f0 t0
f1 t1
A∧B t1 t0 f0 f1
t1 t1 t0 f0 f1
t0 t0 t0 f0 f0
f0 f0 f0 f0 f0
f1 f1 f0 f0 f1
We introduce three different sets of designated
values as follows:
• D
w
1
:= {t1};
• D
w
2
:= {t1, t0};
• D
w
3
:= {t1, t0, f0}.
Based on these sets of designated values, we define
three consequence relations as follows.
Definition 5. For all Γ ∪ {A} ⊆ Form, Γ |=
w
i
A iff
for all four-valued interpretations I
w
4
, I
w
4
(A) ∈ D
w
i
if
I
w
4
(B) ∈ D
w
i
for all B ∈ Γ, where i ∈ {1, 2, 3}.
We have already shown the facts in (Song et al.,
2021) that:
• |=
1
is the weak Kleene logic;
• |=
2
is the classical logic;
• |=
3
is the paraconsistent weak Kleene logic.
3.2 Semantics (II)
If we consider the state of unknown as the second
case, we give the semantics as following:
Definition 6. A two-valued interpretation for the lan-
guage L is a pair hV
s
t
, V
s
a
i, where V
s
t
: Prop → {t, f}
and V
s
a
: Prop → {0, 1}. Valuations V
s
t
, V
s
a
are then ex-
tended to interpretations I
s
t
, I
s
a
by the following condi-
tions.
• I
s
t
(p)=t iff V
s
t
(p)=t
• I
s
a
(p)=1 iff V
s
a
(p)=1
• I
s
t
(∼A)=t iff I
s
t
(A)=f
• I
s
a
(∼A)=1 iff I
s
a
(A)=1
A Many-valued Semantics for Multi-agent System
865
• I
s
t
(A ∧ B)=t iff I
s
t
(A)=t and I
s
t
(B)=t
• I
s
a
(A ∧ B)=1 iff (I
s
t
(A ∧ B)=t and I
s
a
(A)=1 and
I
s
a
(B)=1) or (I
s
t
(A)=f and I
s
a
(A)=1) or (I
s
t
(B)=f
and I
s
a
(B)=1))
The definition of I
s
a
(A ∧B) may seem strange. Ac-
tually, we just consider it as the S5 system of the epis-
temic logic. For example, we consider the I
s
a
(A∧B) =
1 as
a
(A ∧ B) ∨
a
¬(A ∧ B). Therefore there exists
three possible cases in all S5 models that
• A ∧ B and
a
A ∨
a
¬A and
a
B ∨
a
¬B;
• ¬A and
a
A ∨
a
¬A;
• ¬B and
a
B ∨
a
¬B.
which is the same as the definition we showed above.
Also, we can see the semantics more clearly by the
truth table.
Definition 7. A four-valued interpretation of L is
a function I
s
4
: Prop → {t1, t0, f0, f1}. Given a four-
valued interpretation I
s
4
, this is extended to a function
that assigns every formula a truth value by the follow-
ing truth functions:
A ∼ A
t1 f1
t0 f0
f0 t0
f1 t1
A∧B t1 t0 f0 f1
t1 t1 t0 f0 f1
t0 t0 t0 f0 f1
f0 f0 f0 f0 f1
f1 f1 f1 f1 f1
We introduce three different sets of designated
values as follows:
• D
s
1
:= {t1};
• D
s
2
:= {t1, t0};
• D
s
3
:= {t1, t0, f0}.
Based on these sets of designated values, we define
three consequence relations as follows.
Definition 8. For all Γ ∪ {A} ⊆ Form, Γ |=
s
i
A iff for
all four-valued interpretations I
s
4
, I
s
4
(A) ∈ D
i
if I
s
4
(B) ∈
D
s
i
for all B ∈ Γ, where i ∈ {1, 2, 3}.
Then, we can show the facts that:
• |=
s
1
is the strong Kleene logic;
• |=
s
2
is the classical logic;
• |=
s
3
is the logic of paradox.
We first deal with the case in which t1 is the only
designated value. To show the first fact, we prepare a
lemma.
Lemma 1. For all strong Kleene three-valued valua-
tion v
s
3
for L, there is a four-valued valuation v
s
4
such
that for all A ∈ Form, (i) I
s
4
(A) = t1 iff I
s
3
(A) = t, and
(ii) I
s
4
(A) = f1 iff I
s
3
(A) = f.
Proof. Given a three-valued valuation v
s
3
, we define
v
s
4
: Prop → {t1, t0, f0, f1} as follows:
v
s
4
(p)=
t1 v
s
3
(p) = t
f0 v
s
3
(p) = b
f1 v
s
3
(p) = f
Then we prove the desired result by induction on the
complexity of the formula. For the base case, the de-
sired result holds by the definition of v
s
4
. For the in-
duction step, we split the cases depending on the form
of the formula A.
If A is of the form ∼B, then for (i), we have
I
s
4
(A) = t1 iff I
s
4
(∼B)=t1 iff I
s
4
(B)=f1 (by def. of
I
s
4
) iff I
s
3
(B)=f (by IH) iff I
s
3
(∼B)=t (by def. of
I
s
3
) iff I
s
3
(A)=t. For (ii), I
s
4
(A) = f1 iff I
s
4
(∼B)=f1
iff I
s
4
(B)=t1 (by def. of I
s
4
) iff I
s
3
(B)=t (by IH) iff
I
s
3
(∼B)=f (by def. of I
s
3
) iff I
s
3
(A)=f.
If A is of the form B∧C, then for (i), I
s
4
(A)=t1 iff
I
s
4
(B∧C)=t1 iff I
s
4
(B)=t1 and I
s
4
(C)=t1 (by def. of
I
s
4
) iff I
s
3
(B)=t and I
s
3
(C)=t (by IH) iff I
s
3
(B∧C)=t
(by def. of I
s
3
) iff I
s
3
(A)=t. For (ii), I
s
4
(A)=f1 iff
I
s
4
(B∧C)=f1 iff I
s
4
(B)=f1 or I
s
4
(C)=f1(by def. of I
s
4
)
iff I
s
3
(B)=f or I
s
3
(C)=f(by IH) iff I
s
3
(B∧C)=f (by def.
of I
s
3
) iff I
s
3
(A)=f.
The case for disjunction is similar.
We are now ready to prove one of the directions.
Proposition 1. For Γ ∪ {A}⊆Form, if Γ|=
s
1
A then
Γ|=
SK
A.
Proof. Suppose Γ 6|=
SK
A. Then, there is a three-
valued valuation v
s
3
: Prop → {t, b, f} such that
I
s
3
(B) = t for all B ∈ Γ and I
s
3
(A) 6= t. Now, in view
of (i) of Lemma 1, there is a four-valued valuation v
s
4
such that I
s
4
(B)=t1 for all B∈Γ and I
s
4
(A) 6= t1, namely
Γ 6|=
s
1
A, as desired.
For the other direction, we prepare another
lemma.
Lemma 2. For all four-valued valuation v
s
4
for L ,
there is a strong Kleene three-valued valuation v
s
3
such that for all A ∈ Form, (i) I
s
3
(A) = t iff I
s
4
(A) = t1,
and (ii) I
s
3
(A) = f iff I
s
4
(A) = f1.
Proof. Given a four-valued valuation v
s
4
, we define
v
s
3
: Prop → {t, b, f} as follows:
v
s
3
(p)=
t v
s
4
(p) = t1
b v
s
4
(p) = t0 or v
s
4
(p) = f0
f v
s
4
(p) = f1
Then we prove the desired result by induction.
Then, the proof is similar to the above case.
Proposition 2. For Γ ∪ {A}⊆Form, if Γ|=
SK
A then
Γ|=
s
1
A.
ICAART 2022 - 14th International Conference on Agents and Artificial Intelligence
866
Proof. Suppose Γ 6|=
s
1
A. Then, there is a four-
valued valuation v
s
4
: Prop → {t1, t0, f0, f1} such that
I
s
4
(B)=t1 for all B∈Γ and I
s
4
(A) 6= t1. Now, in view
of (i) of Lemma 2, there is a three-valued valuation
v
s
3
such that I
s
3
(B)=t for all B∈Γ and I
s
3
(A)6=t, namely
Γ 6|=
SK
A, as desired.
In view of the above propositions, we obtain the
following.
Theorem 1. For all Γ ∪ {A} ⊆ Form, Γ |=
SK
A iff
Γ |=
s
1
A.
In other words, this semantics is equivalent to the
strong Kleene logic.
Then, consider the case for the logic of paradox, in
which t1, t0 and f0 are taken as designated values. In
fact, the proofs are basically the same with the cases
for the strong Kleene logic.
Theorem 2. For all Γ ∪ {A} ⊆ Form, Γ |=
LP
A iff
Γ |=
s
3
A.
Finally, we consider the case in which t1 and t0
are designated. Actually, the proof is just the same
as |=
w
2
of Semantics (I) which we showed in (Song
et al., 2021). Therefore, we can obtain the following
theorem.
Theorem 3. For all Γ ∪ {A} ⊆ Form, Γ |=
CL
A iff
Γ |=
s
2
A.
4 A MANY-VALUED SEMANTICS
FOR MULTI-AGENT SYSTEM
In the previous section, we give two different seman-
tics for the different readings of unknown. If we con-
sider the two reading of unknown as two different
epistemic states, we can give a more general seman-
tics.
Actually, we can consider the epistemic states as
following:
• The agent knows that A is true or agent knows that
A is false. (I
s
a
(A) = 1 in Semantics (I) or I
w
a
(A) = 1
in Semantic (II))
• The agent doesn’t know the content of A.
(I
w
a
(A) = 0 in Semantics (I))
• The agent considers that A is possibly true and A
is possibly false, i.e., the agent knows the contents
of A but doesn’t know whether A is true or not.
(I
s
a
(A) = 0 in Semantic (II))
Therefore, we combine the two semantics above
to extend the valuation V
a
to three-valued {1, 0.5, 0}
and read the values as following:
• V
a
(p) = 1: Agent a knows the content of p and
whether p is true or not.
• V
a
(p) = 0: Agent a doesn’t know the content of
p.
• V
a
(p) = 0.5: Agent a knows the content of p but
doesn’t know whether p is true or not.
According to the consideration above, first, we
give the semantics for a single agent.
4.1 Many-valued Semantics for Single
Agent
Definition 9. A many-valued semantics for the lan-
guage L is a pair hV
t
, V
a
i, where V
t
: Prop → {t, f}
and V
a
: Prop → {0, 0.5, 1}. Valuations V
t
, V
a
are then
extended to interpretations I
t
, I
a
by the following truth
table. Here, to show the truth table clearly, we write
the valuation I
6
: Prop → {t1, t0.5, t0, f0, f0.5, f1} as a
6-valued logic instead of the pair (I
t
, I
a
).
A ∼ A
t1 f1
t0.5 f0.5
t0 f0
f0 t0
f0.5 t0.5
f1 t
A∧B t1 t0.5 t0 f0 f0.5 f1
t1 t1 t0.5 t0 f0 f0.5 f1
t0.5 t0.5 t0.5 t0 f0 f0.5 f1
t0 t0 t0 t0 f0 f0 f0
f0 f0 f0 f0 f0 f0 f0
f0.5 f0.5 f0.5 f0 f0 f0.5 f1
f1 f1 f1 f0 f0 f1 f1
We introduce three different sets of designated
values as follows:
• D
1
:= {t1}, if we ask the agent that what true
statement it knows;
• D
2
:= {t1, t0.5, t0}, if we ask that what true state-
ments are(may be answered by an omniscient
agent);
• D
3
:= {t1, t0.5, f0.5}, if we ask the agent that
what statement the agent considers that may be
true.
Based on these sets of designated values, we de-
fine the consequence relations as follows.
Definition 10. For all Γ ∪ {A} ⊆ Form, Γ |=
i
A iff for
all four-valued interpretations I
6
, I
6
(A) ∈ D
i
if I
6
(B) ∈
D
i
for all B ∈ Γ, where i ∈ {1, 2, 3}.
Then, we can show the facts that:
• |=
2
is the classical logic;
A Many-valued Semantics for Multi-agent System
867
• |=
3
is the S
f de
logic.
The proof of |=
2
is just the same as above because it is
just the classical logic if we only take care of V
t
and I
t
.
To deal with the case of |=
3
, we prepare two lemmas.
Lemma 3. For all S
f de
valuation v
s f de
for L, there is
a six-valued valuation v
6
such that for all A ∈ Form,
• (i) I
6
(A) = t1 iff I
s f de
(A) = t;
• (ii) I
6
(A) ∈ {t0.5, f0.5} iff I
s f de
(A) = b;
• (iii) I
6
(A) ∈ {t0, f0} iff I
s f de
(A) = n;
• (iv) I
6
(A) = f1 iff I
s f de
(A) = f;
Proof. Given a S
f de
valuation v
s f de
, we define v
6
:
Prop → {t1, t0.5, t0, f0, f0.5, f1} as follows:
v
6
(p)=
t1 v
s f de
(p) = t
f0.5 v
s f de
(p) = b
f0 v
s f de
(p) = n
f1 v
s f de
(p) = f
Then we prove the desired result by induction.
Lemma 4. For all six-valued valuation v
6
for L , there
is a strong Kleene three-valued valuation v
s f de
such
that for all A ∈ Form,
• (i) I
s f de
(A) = t iff I
6
(A) = t1;
• (ii) I
s f de
(A) = b iff I
6
(A) ∈ {t0.5, f0.5};
• (iii) I
s f de
(A) = n iff I
6
(A ∈ {t0, f0};
• (iv) I
s f de
(A) = f iff I
6
(A) = f1;
Proof. Given a six-valued valuation v
6
, we define
v
s f de
: Prop → {t, b, n, f} as follows:
v
f de
(p)=
t v
6
(p) = t
b v
6
(p) = t0.5 or v
6
(p) = f0.5
n v
6
(p) = t0 or v
6
(p) = f0
f v
6
(p) = f1
Then we prove the desired result by induction.
Also, it is easy to see that |=
1
is the logic if we
replace the designate values D of S
f de
by D = {t}.
4.2 Many-valued Semantics for
Multi-agent System
Consider that there are several agents, therefore there
should be several valuations of V
a
. Then we give a
many-valued semantics for multi-agent system.
Definition 11. A many-valued semantics for the lan-
guage L is a pair hV
t
, {V
a
}
a∈Ag
i, where Ag is a
non-empty set of agents, V
t
: Prop → {t, f} and V
a
:
Prop → {0, 0.5, 1}. Valuations V
t
, V
a
are then ex-
tended to interpretations I
t
, I
a
by the following con-
ditions.
• I
t
is the same as the classical logic.
• I
a
(p)=V
a
(p)
• I
a
(∼A)=1 − I
a
(A)
• I
a
(A ∧ B)=1 iff (I
t
(A ∧ B)=t and I
a
(A)=1 and
I
a
(B)=1) or (I
t
(A)=f and I
a
(A)=1 and I
a
(B)6=0)
or (I
t
(B)=f and I
a
(B)=1 and I
a
(A)6=0))
• I
a
(A ∧ B)=0.5 iff (I
t
(A ∧ B)=f and I
a
(A)=0.5
and I
a
(B)=0.5) or (I
t
(A)=t and I
a
(A)=0.5
and I
a
(B)6=0) or (I
t
(B)=t and I
a
(B)=0.5 and
I
a
(A)6=0))
• I
a
(A ∧ B)=0 iff I
a
(A)=0 or I
a
(B)=0
Also, we introduce some different sets of desig-
nated values as follows. To see it clearly, we write the
value as I
t
, I
a
1
, ...I
a
n
.
• D
k
1
:= {t, I
a
1
, ..., I
a
n
: I
a
k
= 1}, if we ask the agent
k that what true statements k knows;
• D
k
2
:= {t, I
a
1
, ..., I
a
n
: I
a
j
∈ {1, 0.5, 0}( j ∈ (1, n))},
if we ask that what true statements are(may be an-
swered by an omniscient agent);
• D
k
3
:= {t, I
a
1
, ..., I
a
n
: I
a
k
∈ {1, 0.5}} ∪
{f, I
a
1
, ..., I
a
n
: I
a
k
= 0.5}, if we ask the agent
k what agent k considers that may be true.
Based on these sets of designated values, we de-
fine three consequence relations as follows.
Definition 12. For all Γ ∪{A} ⊆ Form, Γ |=
k
i
A iff for
all interpretations I, I(A) ∈ D
k
i
if I(B) ∈ D
k
i
for all
B ∈ Γ, where i ∈ {1, 2, 3}.
We can show the facts that |=
k
2
is the classical logic
and |=
k
3
is the S
f de
logic. The proof is the same as case
of a single agent.
5 AGENT COMMUNICATION
We can see that, in the many-valued semantics for
multi-agent system, the valuation shows both the clas-
sical values and the epistemic states of each agent.
In other words, the valuation can be considered as a
model like the Kripke model we use in dynamic epis-
temic logic. Therefore, we can consider several kinds
of valuation change like the dynamic operators to ex-
press the agent communication. First, we give a defi-
nition of consequence relation which is like the epis-
temic logic.
Definition 13. Let Ag be a non-empty set of
agents, Prop be the set of propositions and V =
{V
t
, {V
a
}
a∈Ag
} be the valuation where V
t
: Prop →
{t, f } and V
a
: Prop → {1, 0.5, 0}. Then we can de-
fine the satisfied functions |=
t
a
and |=
mt
a
as following:
V |=
t
a
A iff I
t
(A) = t and I
a
(A) = 1
V |=
mt
a
A iff (I
t
(A) = t and I
a
(A) = 1)
or I
a
(A) = 0.5
ICAART 2022 - 14th International Conference on Agents and Artificial Intelligence
868
Semantically speaking, V |=
t
a
A means that agent
a knows that A is true, and V |=
mt
a
A means that agent
a considers that A may be true. Actually, |=
t
a
stands
for the |=
k
1
and |=
mt
a
stands for the |=
k
3
we showed in
Section 4.
In dynamic epistemic logic, there are several op-
erators to show the communication between agents.
(Van Ditmarsch, 2014) showed the semantics of agent
announcement that one agent tells others some state-
ment instead of the public announcement. (Hatano
et al., 2015) considered the channel communication
as the semi-private announcement. Unlike the stud-
ies of dynamic epistemic logic above, we do not use a
Kripke model so that we cannot show all of the belief
changes. However, it is very simple to show a cer-
tain kind of changes of knowledge by our semantics.
Moreover, we can give some new ideas by consider-
ing two kinds of agent communication: teaching and
asking.
5.1 Teaching
Here, we consider that teaching is the communication
from agent a to a set of agent G that a will tell every
proposition he/ she knows to the member of G. We
write such agent communication by the operator ↓
a
G
.
Semantically speaking, teaching is considered as the
act that the teacher teaches the knowledge to the stu-
dents. If G has only one member, then the act can be
considered as the semi-private announcement. If G is
the set of all agents, then the act can be considered as
the agent announcement.
Definition 14. Let the original model be V . After the
act that agent a teaches the group G, the new valuation
V
↓
a
G
is defined as following:
For all p ∈ Prop,
• If i 6∈ G, then V
↓
a
G
i
(p) = V
i
(p), and
• If i ∈ G, then
V
↓
a
G
i
(p) :=
(
1 if V
a
(p) = 1
V
i
(p) otherwise.
Semantically speaking, the students of a can ac-
quire the knowledge that is known to a, and the epis-
temic states of other agents will not change.
5.2 Asking
Consider the situation that a student asks a teacher
questions. The student possesses several propositions
that he/ she doesn’t know whether true or false, and
if the teacher knows that, the student can obtain the
answer to know that proposition. It seems that ask-
ing is just the opposite of teaching so that the results
that a teaches b and b asks a are the same. However,
actually sometimes the two cases are different for the
epistemic states of teacher may also be changed by
asked questions. For example, a is the teacher and b is
the student, let V
a
(p) = 0 and V
b
(p) = 0.5, b doesn’t
know whether p is true or not, so he/ she will ask
a about p, while the teacher doesn’t know even the
content of p at the moment. Therefore, after asking,
student b cannot obtain an answer, while teacher a be-
comes to know the content of p. Then, we define the
model V
↑
a
b
that show the model after b asks a ques-
tions as following:
Definition 15. For all p ∈ Prop,
• If i 6∈ {a, b} , then V
↑
a
b
i
(p) = V
i
(p);
• If i = a, then
V
↑
a
b
i
(p) :=
(
1 if V
a
(p) = 0.5 and V
b
(p) = 1
V
i
(p) otherwise.
• If i = b, then
V
↑
a
b
b
(p) :=
(
0.5 if V
a
(p) = 0.5 and V
b
(p) = 0
V
b
(p) otherwise.
6 CONCLUSION AND FUTURE
WORK
In this paper, we showed a many-valued semantics to
express the epistemic states in multi-agent system, in-
stead of epistemic logic. Firstly, we introduced two
pair semantics to express the different considerations
of knowledge, and showed that they could be consid-
ered as the two three-valued logic: weak Kleene logic
and strong Kleene logic. Secondly, we gave a new se-
mantics by combining the two semantics and the two
states of unknown, and extended it to express the epis-
temic states of multi agents. Moreover, we gave two
kinds of agent communication,teaching and asking,
and showed the results of them could be different.
There remain several future works.
• In this paper, we ignored the situation of misun-
derstanding, i.e., agent a believes p is true while
in fact p is false. Here, we considered the multi-
agent system as a knowledge system, which we
often used S5 system in the epistemic logic. To
consider a belief system, we may add a new epis-
temic value to express the state of misunderstand-
ing.
• Using our semantics, there can be more kinds
of agent communication besides what we intro-
duced. For example, discussing makes a group of
agents exchange their knowledge, forgetting lets
some agents lose some information they knew be-
fore, etc..
A Many-valued Semantics for Multi-agent System
869
• In this paper, we showed the relations between our
semantics and several many-valued logics. Actu-
ally, we can also find some connection with other
logics. For example, Fan considered a epistemic
operator 4 to express knowing whether in (Fan
et al., 2015), which has the similar meaning of
the value 0.5 in our semantics. In the awareness
logic(Fagin and Halpern, 1987), the epistemic
states were considered as known with awareness,
unknown with awareness and unaware, which had
the similar readings of our semantics. Thus, we
can make use of our semantics in the epistemic
logic and the awareness logic.
ACKNOWLEDGEMENTS
This research is supported by JSPS kaken 17H02258.
REFERENCES
Belnap, N. D. (1977). A useful four-valued logic. In
Modern uses of multiple-valued logic, pages 5–37.
Springer.
Ciucci, D. and Dubois, D. (2012). Three-valued logics for
incomplete information and epistemic logic. In Eu-
ropean Workshop on Logics in Artificial Intelligence,
pages 147–159. Springer.
Fagin, R. and Halpern, J. Y. (1987). Belief, awareness, and
limited reasoning. Artificial intelligence, 34(1):39–76.
Fan, J., Wang, Y., and Van Ditmarsch, H. (2015). Contin-
gency and knowing whether. The Review of Symbolic
Logic, 8(1):75–107.
Ferguson, T. M. (2017). Rivals to belnap-dunn logic on
interlaced trilattices. Studia Logica: An International
Journal for Symbolic Logic, pages 1123–1148.
Hatano, R., Sano, K., and Tojo, S. (2015). Linear al-
gebraic semantics for multi-agent communication.
In Proceedings of the International Conference on
Agents and Artificial Intelligence (ICAART-2015),
pages 174–181. SCITEPRESS.
Kleene, S. C. (1952). Introduction to metamathematics.
North-Holland, Amsterdam.
Omori, H. and Wansing, H. (2017). 40 years of fde: An
introductory overview. Studia Logica, 105(6):1021–
1049.
Song, Y., Omori, H., and Tojo, S. (2021). A two-valued
semantics for infectious logics. In 2021 IEEE 51st In-
ternational Symposium on Multiple-Valued Logic (IS-
MVL), pages 50–55. IEEE.
Szmuc, D. (2019). An epistemic interpretation of paracon-
sistent weak Kleene logic. Logic and Logical Philos-
ophy, 28(2):277–330.
Szmuc, D. and Omori, H. (2018). A Note on Goddard and
Routley’s Significance Logic. The Australasian Jour-
nal of Logic, 15(2):431–448.
Van Ditmarsch, H. (2014). Dynamics of lying. Synthese,
191(5):745–777.
ICAART 2022 - 14th International Conference on Agents and Artificial Intelligence
870
